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<title>Atlas software user guide -- D4</title>
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<h2>D4</h2>
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<i>Last updated: October 8, 2005</i>
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In type D4, the situation is still a little bit more involved than in the
general case of <a href="Deven.html">type D_2m</a> for m > 2. Here the
automorphism group of the Dynkin diagram is the symmetric group on three
letters, acting faithfully on the three outer vertices of the diagram.
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There are a number of other three-element sets on which Out(G), 
G=<b>Spin</b>(8), acts faithfully:
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<li>the set of non-trivial elements of the center of G (
<a href="Deven.html">recall</a> that this center is isomorphic to 
<b>Z</b>/2.<b>Z</b>/2);
</li>
<li>
The set of non-trivial involutions in Out(G) (in other words, the set of
non-trivial <a href="innerclass.html">inner classes</a> for G.)
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<li>
The set of weak real forms whose most split torus has compact rank two
(such real forms have a Lie algebra isomorphic to <b>so</b>(6,2).) This
is less obvious, but easy to see when real forms for the equal rank
inner class are classified in terms of gradings of the root system.
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In this case, there is no way in which the three two-element subgroups
of the center of G can be distinguished. However, true to our principle that
inner classes matter only up to conjugation in Out(G), we make a choice of
one of the three non-trivial inner classes for G, and label it "u"; the
other two are not accessible in the program. Then we label the elements of
the center in such a way that [1,1] is the one that is fixed by the stabilizer
of our chosen inner class; from there on, we proceed exactly as in the
general case for <a href="Deven.html">type D_2m</a>. For consistency, we
denote <b>so</b>(6,2) the real form that is fixed by the stabilizer of our 
chosen inner class, and <b>so</b>*(8)[0,1] and <b>so</b>*(8)[1,0] the two
others, following the same rule as in the general case, even though of
course <b>so</b>(6,2) and <b>so</b>*(8) are isomorphic Lie algebras, and
give rise to isomorphic real forms for the simply connected or adjoint group.
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<p>
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